By the Perron's formula (see [1]} for example), for \( 1\le T\le X \)  we have
\(\mathcal {S}_l(X) = \frac{1}{2\pi i} \int _{1+\epsilon - iT}^{1+\epsilon +iT} D_l(s) \frac{X^s}{s} ds + O\left(\frac{X^{1+\epsilon }}{T}\right).\) 
